Image Processing Reference

In-Depth Information

Definition 9.4.1 Nearness Relation

LethO,

F

ibe perceptual system and let X, Y⊆O.

A set X is near to a set Y within

the perceptual systemhO,

and there are

A∈O
/∼
A
, B∈O
/∼
B
, C∈O
/∼
f
such that A, B⊆C. If a perceptual system is understood,

then we say briefly that a set X is near to a set Y .

F

i(X

./
F

Y ) iff there areA,B⊆

F

and f∈

F

Definition 9.4.2 Weak Nearness Relation

LethO,

ibe a perceptual system and let X, Y⊆O. A set X is weakly near to a set Y

within the perceptual systemhO,

F

F

i(X./
F
Y ) iff there are x∈X and y∈Y and there is

B⊆

F

such that x∼
B
y, i.e., set X is weakly near to a set Y .

TABLE 9.1 Relation Symbols

Symbol Interpretation

O Set of perceptual objects,

X X⊆O, set of sample objects,

F A set of probe functions,

B B⊆F,

< reals,

φ φ∈B, where φ
i
: O−→<, probe function,

∼
B
{(x, y)| f (x) = f (y)∀φ∈B},

'
B
{(x, y)∈O×O|∃φ
i
∈B, φ
i
(x) = φ
i
(y)},

∼
=
B,ε
Perceptual tolerance relation,

x
/∼
B
x
/∼
B
={y∈X | y∼
B
x}(class),

O
/∼
B
O
/∼
B
={x
/∼
B
| x∈O}, quotient set,

./
B
nearness relation symbol,

./
B
weak nearness relation symbol,

./
B,ε

Weak tolerance nearness relation symbol.

Definition 9.4.3 Weak Tolerance Nearness

Let
O,

be a perceptual system and let X, Y⊆O,
∈

F

R

. The set X is perceptually near

to the set Y within the perceptual system
O,

(X ./
F

F

Y ) iff there exists x∈X, y∈Y

such that x
∼
=
B
y. If a perceptual system is understood, then

we say shortly that a set X is perceptually near to a set Y in a weak tolerance sense of

nearness.

and there is a φ∈

F

,
∈

R

Definition 9.4.4 Tolerance Near Sets

LethO,

ibe a perceptual system and let X⊆O. A set X is a tolerance near set iff there is

Y⊆O such that X ./
F

F

Y . The family of near sets of a perceptual systemhO,

F

iis denoted

by Near
F
(O).

9.5

Three Tolerance Near Set-based Nearness Measures for

Image Analysis and Comparison

The term tolerance space was introduced by E.C. Zeeman in 1961 in modelling visual per-

ception with tolerances. Tolerance relations are viewed as good models of how one perceives,

how one sees (Zeeman, 1961). Tolerance relations are also considered as a basis for studying

similarities between objects (Peters, 2008c). The basic idea behind using tolerance classes

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